3 Regular expressions

Size: px
Start display at page:

Download "3 Regular expressions"

Transcription

1 3 Regulr expressions Given n lphet Σ lnguge is set of words L Σ. So fr we were le to descrie lnguges either y using set theory (i.e. enumertion or comprehension) or y n utomton. In this section we shll introduce regulr expressions s n elegnt nd concise wy to descrie lnguges. We shll see tht the lnguges definle y regulr expressions re precisely the sme s those ccepted y deterministic or nondeterministic finite utomt. These lnguges re clled regulr lnguges or (ccording to the Chomsky hierrchy) Type 3 lnguges. As lredy mentioned in the introduction regulr expressions re used to define ptterns in progrms such s grep. grep gets s n rgument regulr expression nd then filters out ll those lines from file which mtch the regulr expression, where mtching mens tht the line contins sustring which is in the lnguge ssigned to the regulr expression. It is interesting to note tht even in the cse when we serch for specific word (this is specil cse of regulr expresion) progrms like grep re more efficient thn nive implementtion of word serch. To find out more out grep hve look t the UNIX mnul pge nd ply round with grep. Note tht the syntx grep uses is slightly different from the one we use here. grep lso use some convenient shorthnds which re not relevnt for theoreticl nlysis of regulr expressions ecuse they do not extend the clss of lnguges. 3.1 Wht re regulr expressions? We ssume s given n lphet Σ (e.g. Σ = {,, c,..., z}) nd define the syntx of regulr expressions (over Σ) 1. is regulr expression. 2. ɛ is regulr expression. 3. For ech x Σ, x is regulr expression. E.g. in the exmple ll smll letters re regulr expression. We use oldfce to emphsize the difference etween the symol nd the regulr expression. 4. If E nd F re regulr expressions then E + F is regulr expression. 5. If E nd F re regulr expressions then EF (i.e. just one fter the other) is regulr expression. 6. If E is regulr expression then E is regulr expression. 7. If E is regulr expression then (E) is regulr expression. These re ll regulr expressions. Here re some exmples for regulr expressions: ɛ hllo hllo + hello h( + e)llo (ɛ + )() (ɛ + ) As in rithmetic they re some conventions how to red regulr expressions: inds stronger then sequence nd +. E.g. we red s ( ). We hve to use prentheses to enforce the other reding (). Sequencing inds stronger thn +. E.g. we red + cd s () + (c). To enforce nother reding we hve to use prentheses s in ( + c)d. 3.2 The mening of regulr expressions We now know wht regulr expressions re ut wht do they men? For this purpose, we shll first define n opertion on lnguges clled the Kleene str. Given lnguge L Σ we define L = {w 0w 1... w n 1 n N i < n.w i L} Intuitively, L contins ll the words which cn e formed y conctenting n ritrry numer of words in L. This includes the empty word since the numer my e 0. As n exmple consider L = {, } {, } : L = {ɛ,,,,,,,... } You should notice tht we use the sme symol s in Σ ut there is sutle difference: Σ is set of symols ut L Σ is set of words. Alterntively (nd more strctly) one my descrie L s the lest lnguge (wrt ) which contins L nd the empty word nd is closed under conctention: w L v L = wv L We now define the semntics of regulr expressions: To ech regulr expression E over Σ we ssign lnguge L(E) Σ. We do this y induction over the definition of the syntx: 1. L( ) = 2. L(ɛ) = {ɛ} 3. L(x) = {x} where x Σ. 4. L(E + F ) = L(E) L(F ) 5. L(EF ) = {wv w L(E) v L(F )} 6. L(E ) = L(E) 7. L((E)) = L(E) 14 13

2 Sutle points: in 1. the symol my e used s regulr expression (s in L( )) or the empty set ( = {}). Similrily, ɛ in 2. my e regulr expression or word, in 6. my e used to construct regulr expressions or it is n opertion on lnguges. Which lterntive we men ecomes only cler from the context, there is no generlly greed mthemticl nottion 1 to mke this difference explicit. Let us now clculte wht the exmples of regulr expressions from the previous section men, i.e. wht re the lnguges they define: ɛ hllo By 2. L(ɛ) = {ɛ} Let s just look t L(h). We know from 3: Hence y 5: L(h) = {h} L() = {} L(h) = {wv w {h} v {}} = {h} Continuing the sme resoning we otin: hllo + hello h( + e)llo L(hllo) = {hllo} From the previous point we know tht: Hence y using 4 we get: Using 3 nd 4 we know Hence using 5 we otin: L(hllo) = {hllo} L(hello) = {hello} L(hllo + hello) = {hllo} {hello}} = {hllo, hello} L( + e) = {, e} L(h( + e)llo) = {uvw u L(h) v L( + e) w L(llo)} = {uvw u {h} v {, e} w {(llo}} = {hllo, hello} 1 This is different in progrmming, e.g. in JAVA we use "... " to signl tht we men things literlly. Let us introduce the following nottion: Now using 6 we know tht w i = ww... w }{{} i times L( ) = {w 0w 1... w n 1 n N i < n.w i L()} = {w 0w 1... w n 1 n N i < n.w i {}} = { n n N} nd hence using 5 we conclude L( ) = {uv u L( ) v L( )} = {uv u { n n N} v { m m N}} = { n m m, n N} I.e. L( ) is the set of ll words which strt with (possily empty) sequence of s followed y (possily empty) sequence of s. (ɛ + )() (ɛ + ) Let s nlyze the prts: Hence, we hve L(ɛ + ) = {ɛ, } L(() ) = { i i N} L(ɛ + ) = {ɛ, } L((ɛ + )() (ɛ + )) = {u() i v u {ɛ, } i N v {ɛ, } In english: L((ɛ + )() (ɛ + )) is the set of (possily empty) sequences of interchnging s nd s. 3.3 Trnslting regulr expressions to NFAs Theorem 3.1 For ech regulr expression E we cn construct NFA N(E) s.t. L(N(E)) = L(E), i.e. the utomton ccepts the lnguge descried y the regulr expression. Proof: We do this gin y induction on the syntx of regulr expressions: 1. N( ): 16 15

3 x N(0) N(x) which will reject everything (it hs got no finl sttes) nd hence L(N( )) = = L( ) 2. N(ɛ): This utomton only ccepts the word x, hence: L(N(x)) = {x} = L(x) 4. N(E + F ): We merge the digrms for N(E) nd N(F ) into one: N(E) N(F) N( ε) I.e. given N(E+F) N(E) = (Q E, Σ, δ E, S E, F E) N(F ) = (Q F, Σ, δ F, S F, F F ) This utomton ccepts the empty word ut rejects everything else, hence: Now we use the disjoint union opertion on sets (see the MCS lecture L(N(ɛ)) = {ɛ} = L(ɛ) 3. N(x): 18 17

4 notes [Alt01], section 4.1) Q E+F = Q E + Q F δ E+F ((0, q), x) = {(0, q ) q δ E (q, x)} δ E+F ((1, q)), x = {(1, q ) q δ F (q, x)} S E+F = S E + S F F E+F = F E + F F N(E + F ) = (Q E+F, Σ, δ E+F, S E+F, F E+F ) The disjoint union just signls tht we re not going to identify sttes, even if they ccidently hppen to hve the sme nme. Just thinking of the gme with mrkers you should e le to convince yourself tht L(N(E + F )) = L(N(E)) L(N(F )) Moreover to show tht we re llowed to ssume tht L(N(E + F )) = L(E + F ) L(N(E)) = L(E) L(N(F )) = L(F ) tht s wht is ment y induction over the syntx of regulr expressions. Now putting everything together: 5. N(EF ): L(N(E + F )) = L(N(E)) L(N(F )) = L(E) L(F ) = L(E + F ) We wnt to put the two utomt N(E) nd N(F ) in series. We do this y connecting the finl sttes of N(E) with the initil sttes of N(F ) in wy explined elow. In this digrm I only depicted one initil nd one finl stte of ech of the utomt lthough they my e severl of them. Here is how we construct N(EF ) from N(E) nd N(F ): N(E) = (Q E, Σ, δ E, S E, F E) N(F ) = (Q F, Σ, δ F, S F, F F ) The sttes of N(EF ) re the disjoint union of the sttes of N(E) nd N(F ): Q EF = Q E + Q F The trnsition function of N(EF ) contins ll the trnsitions of N(E) nd N(F ) (s for N(E + F )) nd for ech stte q of N(E) which hs trnsition to finl stte of N(E) we dd trnsition with the sme lel to ll the initil sttes of N(F ). δ EF ((0, q), x) = {(0, q ) q δ E(q, x)} {(1, q ) q.q δ E(q, x) q S E} δ EF ((1, q)) = {(1, q ) q δ F (q))} The initil sttes of N(EF ) re the initil sttes of N(E), nd the initil sttes of N(F ) if there is n initil stte of N(E) which is lso finl stte. S EF = {(0, q) q S E} {(1, q) q S F S E F E } The finl sttes of N(EF ) re the finl sttes of N(F ). We now set F EF = {(1, q) q F F } N(EF ) = (Q EF, Σ, δ EF, S EF, Z EF ) I hope tht you re le to convince yourself tht nd hence we cn reson L(N(EF )) = {uv u L(N(E)) v L(N(F )) N(E) N(F) L(N(EF )) = {uv u L(N(E)) v L(N(F )) = {uv u L(E) v L(F ) 6. N(E ): = L(EF ) We construct N(E ) from N(E) y merging initil nd finl sttes of N(E) in wy similr to the previous construction nd we dd new stte which is initil nd finl. N(EF) 20 19

5 * since we cn run through the utomton n ritrry numer of times. The new stte llows us lso to ccept the empty sequence. Hence: L(N(E )) = {w 0w 1... w n 1 n N i < n.w i L(N(E))} = L(N(E)) = L(E) = L(E ) N(E) N(E*) 7. N((E)) = N(E) I.e. using rckets does not chnge nything. As n exmple we construct N( ). First we construct N(): Given we construct N(E ). N(E) = (Q E, Σ, δ E, S E, F E ) We dd one extr stte : Q E = Q E + { } N E inherits ll trnsitions form N E nd for ech stte which hs n rrow to the finl stte lelled x we lso dd n rrow to ll the initil sttes lelled x. N() Now we hve to pply the -construction nd we otin: δ E ((0, q), x) ={(0, q ) q δ E (q, x)} {(0, q ) δ E (q, x) F E q S E } The initil sttes of N(E ) re the initil sttes of N(E) nd : S E = {(0, q) q S E } {(1, )} The finl sttes of N E re the finl sttes of N E nd : F E = {(0, q) q F E } {(1, )} We define We clim tht N(E ) = (Q E, Σ, δ E, S E, F E ) N(*) L(N(E )) = {w 0 w 1... w n 1 n N i < n.w i L(N(E))} N( ) is just the sme nd we get 22 21

6 N(*) N(*) nd now we hve to serilize the two utomt nd we get: 1. L is given y regulr expression. 2. L is the lnguge ccepted y n NFA. 3. L is the lnguge cceped y DFA. Proof: We hve tht 1. = 2 y theorem 3.1. We know tht 2. = 3. y2.2 nd 3. = 1. y 3.2. As indicted in the introduction: the lnguges which re chrcterized y ny of the three equivlent conditions re clled regulr lnguges or type-3- lnguges. N(**) Now, you my oserve tht this utomton, though correct, is unnecessry complicted, since we could hve just used However, we shll not e concerned with minimlity t the moment. 3.4 Summing up... From the previous section we know tht lnguge given y regulr expression is lso recognized y NFA. Wht out the other wy: Cn lnguge recognized y finite utomton (DFA or NFA) lso e descried y regulr expression? The nswer is yes: Theorem 3.2 (Theorem 3.4, pge 91) Given DFA A there is regulr expression R(A) which recognizes the sme lnguge L(A) = L(R(A)). We omit the proof (which cn e found in the [HMU01] on pp.91-93). However, we conclude: Corollry 3.3 Given lnguge L Σ the following is equivlent: 24 23

AUTOMATA AND LANGUAGES. Definition 1.5: Finite Automaton

AUTOMATA AND LANGUAGES. Definition 1.5: Finite Automaton 25. Finite Automt AUTOMATA AND LANGUAGES A system of computtion tht only hs finite numer of possile sttes cn e modeled using finite utomton A finite utomton is often illustrted s stte digrm d d d. d q

More information

Designing finite automata II

Designing finite automata II Designing finite utomt II Prolem: Design DFA A such tht L(A) consists of ll strings of nd which re of length 3n, for n = 0, 1, 2, (1) Determine wht to rememer out the input string Assign stte to ech of

More information

Lecture 08: Feb. 08, 2019

Lecture 08: Feb. 08, 2019 4CS4-6:Theory of Computtion(Closure on Reg. Lngs., regex to NDFA, DFA to regex) Prof. K.R. Chowdhry Lecture 08: Fe. 08, 2019 : Professor of CS Disclimer: These notes hve not een sujected to the usul scrutiny

More information

Anatomy of a Deterministic Finite Automaton. Deterministic Finite Automata. A machine so simple that you can understand it in less than one minute

Anatomy of a Deterministic Finite Automaton. Deterministic Finite Automata. A machine so simple that you can understand it in less than one minute Victor Admchik Dnny Sletor Gret Theoreticl Ides In Computer Science CS 5-25 Spring 2 Lecture 2 Mr 3, 2 Crnegie Mellon University Deterministic Finite Automt Finite Automt A mchine so simple tht you cn

More information

1 Nondeterministic Finite Automata

1 Nondeterministic Finite Automata 1 Nondeterministic Finite Automt Suppose in life, whenever you hd choice, you could try oth possiilities nd live your life. At the end, you would go ck nd choose the one tht worked out the est. Then you

More information

CHAPTER 1 Regular Languages. Contents

CHAPTER 1 Regular Languages. Contents Finite Automt (FA or DFA) CHAPTE 1 egulr Lnguges Contents definitions, exmples, designing, regulr opertions Non-deterministic Finite Automt (NFA) definitions, euivlence of NFAs nd DFAs, closure under regulr

More information

Fundamentals of Computer Science

Fundamentals of Computer Science Fundmentls of Computer Science Chpter 3: NFA nd DFA equivlence Regulr expressions Henrik Björklund Umeå University Jnury 23, 2014 NFA nd DFA equivlence As we shll see, it turns out tht NFA nd DFA re equivlent,

More information

CS 301. Lecture 04 Regular Expressions. Stephen Checkoway. January 29, 2018

CS 301. Lecture 04 Regular Expressions. Stephen Checkoway. January 29, 2018 CS 301 Lecture 04 Regulr Expressions Stephen Checkowy Jnury 29, 2018 1 / 35 Review from lst time NFA N = (Q, Σ, δ, q 0, F ) where δ Q Σ P (Q) mps stte nd n lphet symol (or ) to set of sttes We run n NFA

More information

Finite Automata-cont d

Finite Automata-cont d Automt Theory nd Forml Lnguges Professor Leslie Lnder Lecture # 6 Finite Automt-cont d The Pumping Lemm WEB SITE: http://ingwe.inghmton.edu/ ~lnder/cs573.html Septemer 18, 2000 Exmple 1 Consider L = {ww

More information

Convert the NFA into DFA

Convert the NFA into DFA Convert the NF into F For ech NF we cn find F ccepting the sme lnguge. The numer of sttes of the F could e exponentil in the numer of sttes of the NF, ut in prctice this worst cse occurs rrely. lgorithm:

More information

Homework 3 Solutions

Homework 3 Solutions CS 341: Foundtions of Computer Science II Prof. Mrvin Nkym Homework 3 Solutions 1. Give NFAs with the specified numer of sttes recognizing ech of the following lnguges. In ll cses, the lphet is Σ = {,1}.

More information

Finite-State Automata: Recap

Finite-State Automata: Recap Finite-Stte Automt: Recp Deepk D Souz Deprtment of Computer Science nd Automtion Indin Institute of Science, Bnglore. 09 August 2016 Outline 1 Introduction 2 Forml Definitions nd Nottion 3 Closure under

More information

Theory of Computation Regular Languages. (NTU EE) Regular Languages Fall / 38

Theory of Computation Regular Languages. (NTU EE) Regular Languages Fall / 38 Theory of Computtion Regulr Lnguges (NTU EE) Regulr Lnguges Fll 2017 1 / 38 Schemtic of Finite Automt control 0 0 1 0 1 1 1 0 Figure: Schemtic of Finite Automt A finite utomton hs finite set of control

More information

Harvard University Computer Science 121 Midterm October 23, 2012

Harvard University Computer Science 121 Midterm October 23, 2012 Hrvrd University Computer Science 121 Midterm Octoer 23, 2012 This is closed-ook exmintion. You my use ny result from lecture, Sipser, prolem sets, or section, s long s you quote it clerly. The lphet is

More information

Intermediate Math Circles Wednesday, November 14, 2018 Finite Automata II. Nickolas Rollick a b b. a b 4

Intermediate Math Circles Wednesday, November 14, 2018 Finite Automata II. Nickolas Rollick a b b. a b 4 Intermedite Mth Circles Wednesdy, Novemer 14, 2018 Finite Automt II Nickols Rollick nrollick@uwterloo.c Regulr Lnguges Lst time, we were introduced to the ide of DFA (deterministic finite utomton), one

More information

NFAs and Regular Expressions. NFA-ε, continued. Recall. Last class: Today: Fun:

NFAs and Regular Expressions. NFA-ε, continued. Recall. Last class: Today: Fun: CMPU 240 Lnguge Theory nd Computtion Spring 2019 NFAs nd Regulr Expressions Lst clss: Introduced nondeterministic finite utomt with -trnsitions Tody: Prove n NFA- is no more powerful thn n NFA Introduce

More information

Theory of Computation Regular Languages

Theory of Computation Regular Languages Theory of Computtion Regulr Lnguges Bow-Yw Wng Acdemi Sinic Spring 2012 Bow-Yw Wng (Acdemi Sinic) Regulr Lnguges Spring 2012 1 / 38 Schemtic of Finite Automt control 0 0 1 0 1 1 1 0 Figure: Schemtic of

More information

Chapter Five: Nondeterministic Finite Automata. Formal Language, chapter 5, slide 1

Chapter Five: Nondeterministic Finite Automata. Formal Language, chapter 5, slide 1 Chpter Five: Nondeterministic Finite Automt Forml Lnguge, chpter 5, slide 1 1 A DFA hs exctly one trnsition from every stte on every symol in the lphet. By relxing this requirement we get relted ut more

More information

CHAPTER 1 Regular Languages. Contents. definitions, examples, designing, regular operations. Non-deterministic Finite Automata (NFA)

CHAPTER 1 Regular Languages. Contents. definitions, examples, designing, regular operations. Non-deterministic Finite Automata (NFA) Finite Automt (FA or DFA) CHAPTER Regulr Lnguges Contents definitions, exmples, designing, regulr opertions Non-deterministic Finite Automt (NFA) definitions, equivlence of NFAs DFAs, closure under regulr

More information

Deterministic Finite Automata

Deterministic Finite Automata Finite Automt Deterministic Finite Automt H. Geuvers nd J. Rot Institute for Computing nd Informtion Sciences Version: fll 2016 J. Rot Version: fll 2016 Tlen en Automten 1 / 21 Outline Finite Automt Finite

More information

NFAs continued, Closure Properties of Regular Languages

NFAs continued, Closure Properties of Regular Languages Algorithms & Models of Computtion CS/ECE 374, Fll 2017 NFAs continued, Closure Properties of Regulr Lnguges Lecture 5 Tuesdy, Septemer 12, 2017 Sriel Hr-Peled (UIUC) CS374 1 Fll 2017 1 / 31 Regulr Lnguges,

More information

CMPSCI 250: Introduction to Computation. Lecture #31: What DFA s Can and Can t Do David Mix Barrington 9 April 2014

CMPSCI 250: Introduction to Computation. Lecture #31: What DFA s Can and Can t Do David Mix Barrington 9 April 2014 CMPSCI 250: Introduction to Computtion Lecture #31: Wht DFA s Cn nd Cn t Do Dvid Mix Brrington 9 April 2014 Wht DFA s Cn nd Cn t Do Deterministic Finite Automt Forml Definition of DFA s Exmples of DFA

More information

Homework 4. 0 ε 0. (00) ε 0 ε 0 (00) (11) CS 341: Foundations of Computer Science II Prof. Marvin Nakayama

Homework 4. 0 ε 0. (00) ε 0 ε 0 (00) (11) CS 341: Foundations of Computer Science II Prof. Marvin Nakayama CS 341: Foundtions of Computer Science II Prof. Mrvin Nkym Homework 4 1. UsetheproceduredescriedinLemm1.55toconverttheregulrexpression(((00) (11)) 01) into n NFA. Answer: 0 0 1 1 00 0 0 11 1 1 01 0 1 (00)

More information

Assignment 1 Automata, Languages, and Computability. 1 Finite State Automata and Regular Languages

Assignment 1 Automata, Languages, and Computability. 1 Finite State Automata and Regular Languages Deprtment of Computer Science, Austrlin Ntionl University COMP2600 Forml Methods for Softwre Engineering Semester 2, 206 Assignment Automt, Lnguges, nd Computility Smple Solutions Finite Stte Automt nd

More information

CS103B Handout 18 Winter 2007 February 28, 2007 Finite Automata

CS103B Handout 18 Winter 2007 February 28, 2007 Finite Automata CS103B ndout 18 Winter 2007 Ferury 28, 2007 Finite Automt Initil text y Mggie Johnson. Introduction Severl childrens gmes fit the following description: Pieces re set up on plying ord; dice re thrown or

More information

Let's start with an example:

Let's start with an example: Finite Automt Let's strt with n exmple: Here you see leled circles tht re sttes, nd leled rrows tht re trnsitions. One of the sttes is mrked "strt". One of the sttes hs doule circle; this is terminl stte

More information

Non-deterministic Finite Automata

Non-deterministic Finite Automata Non-deterministic Finite Automt Eliminting non-determinism Rdoud University Nijmegen Non-deterministic Finite Automt H. Geuvers nd T. vn Lrhoven Institute for Computing nd Informtion Sciences Intelligent

More information

CMSC 330: Organization of Programming Languages

CMSC 330: Organization of Programming Languages CMSC 330: Orgniztion of Progrmming Lnguges Finite Automt 2 CMSC 330 1 Types of Finite Automt Deterministic Finite Automt (DFA) Exctly one sequence of steps for ech string All exmples so fr Nondeterministic

More information

Lecture 09: Myhill-Nerode Theorem

Lecture 09: Myhill-Nerode Theorem CS 373: Theory of Computtion Mdhusudn Prthsrthy Lecture 09: Myhill-Nerode Theorem 16 Ferury 2010 In this lecture, we will see tht every lnguge hs unique miniml DFA We will see this fct from two perspectives

More information

5. (±±) Λ = fw j w is string of even lengthg [ 00 = f11,00g 7. (11 [ 00)± Λ = fw j w egins with either 11 or 00g 8. (0 [ ffl)1 Λ = 01 Λ [ 1 Λ 9.

5. (±±) Λ = fw j w is string of even lengthg [ 00 = f11,00g 7. (11 [ 00)± Λ = fw j w egins with either 11 or 00g 8. (0 [ ffl)1 Λ = 01 Λ [ 1 Λ 9. Regulr Expressions, Pumping Lemm, Right Liner Grmmrs Ling 106 Mrch 25, 2002 1 Regulr Expressions A regulr expression descries or genertes lnguge: it is kind of shorthnd for listing the memers of lnguge.

More information

Minimal DFA. minimal DFA for L starting from any other

Minimal DFA. minimal DFA for L starting from any other Miniml DFA Among the mny DFAs ccepting the sme regulr lnguge L, there is exctly one (up to renming of sttes) which hs the smllest possile numer of sttes. Moreover, it is possile to otin tht miniml DFA

More information

CS 373, Spring Solutions to Mock midterm 1 (Based on first midterm in CS 273, Fall 2008.)

CS 373, Spring Solutions to Mock midterm 1 (Based on first midterm in CS 273, Fall 2008.) CS 373, Spring 29. Solutions to Mock midterm (sed on first midterm in CS 273, Fll 28.) Prolem : Short nswer (8 points) The nswers to these prolems should e short nd not complicted. () If n NF M ccepts

More information

Finite Automata. Informatics 2A: Lecture 3. John Longley. 22 September School of Informatics University of Edinburgh

Finite Automata. Informatics 2A: Lecture 3. John Longley. 22 September School of Informatics University of Edinburgh Lnguges nd Automt Finite Automt Informtics 2A: Lecture 3 John Longley School of Informtics University of Edinburgh jrl@inf.ed.c.uk 22 September 2017 1 / 30 Lnguges nd Automt 1 Lnguges nd Automt Wht is

More information

Formal languages, automata, and theory of computation

Formal languages, automata, and theory of computation Mälrdlen University TEN1 DVA337 2015 School of Innovtion, Design nd Engineering Forml lnguges, utomt, nd theory of computtion Thursdy, Novemer 5, 14:10-18:30 Techer: Dniel Hedin, phone 021-107052 The exm

More information

Converting Regular Expressions to Discrete Finite Automata: A Tutorial

Converting Regular Expressions to Discrete Finite Automata: A Tutorial Converting Regulr Expressions to Discrete Finite Automt: A Tutoril Dvid Christinsen 2013-01-03 This is tutoril on how to convert regulr expressions to nondeterministic finite utomt (NFA) nd how to convert

More information

Regular Expressions (RE) Regular Expressions (RE) Regular Expressions (RE) Regular Expressions (RE) Kleene-*

Regular Expressions (RE) Regular Expressions (RE) Regular Expressions (RE) Regular Expressions (RE) Kleene-* Regulr Expressions (RE) Regulr Expressions (RE) Empty set F A RE denotes the empty set Opertion Nottion Lnguge UNIX Empty string A RE denotes the set {} Alterntion R +r L(r ) L(r ) r r Symol Alterntion

More information

Chapter 2 Finite Automata

Chapter 2 Finite Automata Chpter 2 Finite Automt 28 2.1 Introduction Finite utomt: first model of the notion of effective procedure. (They lso hve mny other pplictions). The concept of finite utomton cn e derived y exmining wht

More information

1. For each of the following theorems, give a two or three sentence sketch of how the proof goes or why it is not true.

1. For each of the following theorems, give a two or three sentence sketch of how the proof goes or why it is not true. York University CSE 2 Unit 3. DFA Clsses Converting etween DFA, NFA, Regulr Expressions, nd Extended Regulr Expressions Instructor: Jeff Edmonds Don t chet y looking t these nswers premturely.. For ech

More information

Formal Languages and Automata

Formal Languages and Automata Moile Computing nd Softwre Engineering p. 1/5 Forml Lnguges nd Automt Chpter 2 Finite Automt Chun-Ming Liu cmliu@csie.ntut.edu.tw Deprtment of Computer Science nd Informtion Engineering Ntionl Tipei University

More information

NFAs continued, Closure Properties of Regular Languages

NFAs continued, Closure Properties of Regular Languages lgorithms & Models of omputtion S/EE 374, Spring 209 NFs continued, losure Properties of Regulr Lnguges Lecture 5 Tuesdy, Jnury 29, 209 Regulr Lnguges, DFs, NFs Lnguges ccepted y DFs, NFs, nd regulr expressions

More information

Non-deterministic Finite Automata

Non-deterministic Finite Automata Non-deterministic Finite Automt From Regulr Expressions to NFA- Eliminting non-determinism Rdoud University Nijmegen Non-deterministic Finite Automt H. Geuvers nd J. Rot Institute for Computing nd Informtion

More information

Automata Theory 101. Introduction. Outline. Introduction Finite Automata Regular Expressions ω-automata. Ralf Huuck.

Automata Theory 101. Introduction. Outline. Introduction Finite Automata Regular Expressions ω-automata. Ralf Huuck. Outline Automt Theory 101 Rlf Huuck Introduction Finite Automt Regulr Expressions ω-automt Session 1 2006 Rlf Huuck 1 Session 1 2006 Rlf Huuck 2 Acknowledgement Some slides re sed on Wolfgng Thoms excellent

More information

Nondeterminism and Nodeterministic Automata

Nondeterminism and Nodeterministic Automata Nondeterminism nd Nodeterministic Automt 61 Nondeterminism nd Nondeterministic Automt The computtionl mchine models tht we lerned in the clss re deterministic in the sense tht the next move is uniquely

More information

Worked out examples Finite Automata

Worked out examples Finite Automata Worked out exmples Finite Automt Exmple Design Finite Stte Automton which reds inry string nd ccepts only those tht end with. Since we re in the topic of Non Deterministic Finite Automt (NFA), we will

More information

CISC 4090 Theory of Computation

CISC 4090 Theory of Computation 9/6/28 Stereotypicl computer CISC 49 Theory of Computtion Finite stte mchines & Regulr lnguges Professor Dniel Leeds dleeds@fordhm.edu JMH 332 Centrl processing unit (CPU) performs ll the instructions

More information

NFA DFA Example 3 CMSC 330: Organization of Programming Languages. Equivalence of DFAs and NFAs. Equivalence of DFAs and NFAs (cont.

NFA DFA Example 3 CMSC 330: Organization of Programming Languages. Equivalence of DFAs and NFAs. Equivalence of DFAs and NFAs (cont. NFA DFA Exmple 3 CMSC 330: Orgniztion of Progrmming Lnguges NFA {B,D,E {A,E {C,D {E Finite Automt, con't. R = { {A,E, {B,D,E, {C,D, {E 2 Equivlence of DFAs nd NFAs Any string from {A to either {D or {CD

More information

Types of Finite Automata. CMSC 330: Organization of Programming Languages. Comparing DFAs and NFAs. NFA for (a b)*abb.

Types of Finite Automata. CMSC 330: Organization of Programming Languages. Comparing DFAs and NFAs. NFA for (a b)*abb. CMSC 330: Orgniztion of Progrmming Lnguges Finite Automt 2 Types of Finite Automt Deterministic Finite Automt () Exctly one sequence of steps for ech string All exmples so fr Nondeterministic Finite Automt

More information

Types of Finite Automata. CMSC 330: Organization of Programming Languages. Comparing DFAs and NFAs. Comparing DFAs and NFAs (cont.) Finite Automata 2

Types of Finite Automata. CMSC 330: Organization of Programming Languages. Comparing DFAs and NFAs. Comparing DFAs and NFAs (cont.) Finite Automata 2 CMSC 330: Orgniztion of Progrmming Lnguges Finite Automt 2 Types of Finite Automt Deterministic Finite Automt () Exctly one sequence of steps for ech string All exmples so fr Nondeterministic Finite Automt

More information

Regular expressions, Finite Automata, transition graphs are all the same!!

Regular expressions, Finite Automata, transition graphs are all the same!! CSI 3104 /Winter 2011: Introduction to Forml Lnguges Chpter 7: Kleene s Theorem Chpter 7: Kleene s Theorem Regulr expressions, Finite Automt, trnsition grphs re ll the sme!! Dr. Neji Zgui CSI3104-W11 1

More information

Closure Properties of Regular Languages

Closure Properties of Regular Languages Closure Properties of Regulr Lnguges Regulr lnguges re closed under mny set opertions. Let L 1 nd L 2 e regulr lnguges. (1) L 1 L 2 (the union) is regulr. (2) L 1 L 2 (the conctention) is regulr. (3) L

More information

The University of Nottingham SCHOOL OF COMPUTER SCIENCE A LEVEL 2 MODULE, SPRING SEMESTER LANGUAGES AND COMPUTATION ANSWERS

The University of Nottingham SCHOOL OF COMPUTER SCIENCE A LEVEL 2 MODULE, SPRING SEMESTER LANGUAGES AND COMPUTATION ANSWERS The University of Nottinghm SCHOOL OF COMPUTER SCIENCE LEVEL 2 MODULE, SPRING SEMESTER 2016 2017 LNGUGES ND COMPUTTION NSWERS Time llowed TWO hours Cndidtes my complete the front cover of their nswer ook

More information

Machines and their languages (G51MAL) Lecture notes Spring 2003

Machines and their languages (G51MAL) Lecture notes Spring 2003 Mchines nd their lnguges (G51MAL) Lecture notes Spring 2003 Contents Thorsten Altenkirch April 23, 2004 1 Introduction 3 1.1 Exmples on syntx......................... 3 1.2 Wht is this course bout?.....................

More information

Scanner. Specifying patterns. Specifying patterns. Operations on languages. A scanner must recognize the units of syntax Some parts are easy:

Scanner. Specifying patterns. Specifying patterns. Operations on languages. A scanner must recognize the units of syntax Some parts are easy: Scnner Specifying ptterns source code tokens scnner prser IR A scnner must recognize the units of syntx Some prts re esy: errors mps chrcters into tokens the sic unit of syntx x = x + y; ecomes

More information

Finite Automata. Informatics 2A: Lecture 3. Mary Cryan. 21 September School of Informatics University of Edinburgh

Finite Automata. Informatics 2A: Lecture 3. Mary Cryan. 21 September School of Informatics University of Edinburgh Finite Automt Informtics 2A: Lecture 3 Mry Cryn School of Informtics University of Edinburgh mcryn@inf.ed.c.uk 21 September 2018 1 / 30 Lnguges nd Automt Wht is lnguge? Finite utomt: recp Some forml definitions

More information

1 From NFA to regular expression

1 From NFA to regular expression Note 1: How to convert DFA/NFA to regulr expression Version: 1.0 S/EE 374, Fll 2017 Septemer 11, 2017 In this note, we show tht ny DFA cn e converted into regulr expression. Our construction would work

More information

80 CHAPTER 2. DFA S, NFA S, REGULAR LANGUAGES. 2.6 Finite State Automata With Output: Transducers

80 CHAPTER 2. DFA S, NFA S, REGULAR LANGUAGES. 2.6 Finite State Automata With Output: Transducers 80 CHAPTER 2. DFA S, NFA S, REGULAR LANGUAGES 2.6 Finite Stte Automt With Output: Trnsducers So fr, we hve only considered utomt tht recognize lnguges, i.e., utomt tht do not produce ny output on ny input

More information

Java II Finite Automata I

Java II Finite Automata I Jv II Finite Automt I Bernd Kiefer Bernd.Kiefer@dfki.de Deutsches Forschungszentrum für künstliche Intelligenz Finite Automt I p.1/13 Processing Regulr Expressions We lredy lerned out Jv s regulr expression

More information

ɛ-closure, Kleene s Theorem,

ɛ-closure, Kleene s Theorem, DEGefW5wiGH2XgYMEzUKjEmtCDUsRQ4d 1 A nice pper relevnt to this course is titled The Glory of the Pst 2 NICTA Resercher, Adjunct t the Austrlin Ntionl University nd Griffith University ɛ-closure, Kleene

More information

Name Ima Sample ASU ID

Name Ima Sample ASU ID Nme Im Smple ASU ID 2468024680 CSE 355 Test 1, Fll 2016 30 Septemer 2016, 8:35-9:25.m., LSA 191 Regrding of Midterms If you elieve tht your grde hs not een dded up correctly, return the entire pper to

More information

1. For each of the following theorems, give a two or three sentence sketch of how the proof goes or why it is not true.

1. For each of the following theorems, give a two or three sentence sketch of how the proof goes or why it is not true. York University CSE 2 Unit 3. DFA Clsses Converting etween DFA, NFA, Regulr Expressions, nd Extended Regulr Expressions Instructor: Jeff Edmonds Don t chet y looking t these nswers premturely.. For ech

More information

In-depth introduction to main models, concepts of theory of computation:

In-depth introduction to main models, concepts of theory of computation: CMPSCI601: Introduction Lecture 1 In-depth introduction to min models, concepts of theory of computtion: Computility: wht cn e computed in principle Logic: how cn we express our requirements Complexity:

More information

CSCI 340: Computational Models. Kleene s Theorem. Department of Computer Science

CSCI 340: Computational Models. Kleene s Theorem. Department of Computer Science CSCI 340: Computtionl Models Kleene s Theorem Chpter 7 Deprtment of Computer Science Unifiction In 1954, Kleene presented (nd proved) theorem which (in our version) sttes tht if lnguge cn e defined y ny

More information

CS375: Logic and Theory of Computing

CS375: Logic and Theory of Computing CS375: Logic nd Theory of Computing Fuhu (Frnk) Cheng Deprtment of Computer Science University of Kentucky 1 Tble of Contents: Week 1: Preliminries (set lgebr, reltions, functions) (red Chpters 1-4) Weeks

More information

Compiler Design. Fall Lexical Analysis. Sample Exercises and Solutions. Prof. Pedro C. Diniz

Compiler Design. Fall Lexical Analysis. Sample Exercises and Solutions. Prof. Pedro C. Diniz University of Southern Cliforni Computer Science Deprtment Compiler Design Fll Lexicl Anlysis Smple Exercises nd Solutions Prof. Pedro C. Diniz USC / Informtion Sciences Institute 4676 Admirlty Wy, Suite

More information

First Midterm Examination

First Midterm Examination 24-25 Fll Semester First Midterm Exmintion ) Give the stte digrm of DFA tht recognizes the lnguge A over lphet Σ = {, } where A = {w w contins or } 2) The following DFA recognizes the lnguge B over lphet

More information

Parse trees, ambiguity, and Chomsky normal form

Parse trees, ambiguity, and Chomsky normal form Prse trees, miguity, nd Chomsky norml form In this lecture we will discuss few importnt notions connected with contextfree grmmrs, including prse trees, miguity, nd specil form for context-free grmmrs

More information

Non-Deterministic Finite Automata. Fall 2018 Costas Busch - RPI 1

Non-Deterministic Finite Automata. Fall 2018 Costas Busch - RPI 1 Non-Deterministic Finite Automt Fll 2018 Costs Busch - RPI 1 Nondeterministic Finite Automton (NFA) Alphbet ={} q q2 1 q 0 q 3 Fll 2018 Costs Busch - RPI 2 Nondeterministic Finite Automton (NFA) Alphbet

More information

Talen en Automaten Test 1, Mon 7 th Dec, h45 17h30

Talen en Automaten Test 1, Mon 7 th Dec, h45 17h30 Tlen en Automten Test 1, Mon 7 th Dec, 2015 15h45 17h30 This test consists of four exercises over 5 pges. Explin your pproch, nd write your nswer to ech exercise on seprte pge. You cn score mximum of 100

More information

CS 311 Homework 3 due 16:30, Thursday, 14 th October 2010

CS 311 Homework 3 due 16:30, Thursday, 14 th October 2010 CS 311 Homework 3 due 16:30, Thursdy, 14 th Octoer 2010 Homework must e sumitted on pper, in clss. Question 1. [15 pts.; 5 pts. ech] Drw stte digrms for NFAs recognizing the following lnguges:. L = {w

More information

Coalgebra, Lecture 15: Equations for Deterministic Automata

Coalgebra, Lecture 15: Equations for Deterministic Automata Colger, Lecture 15: Equtions for Deterministic Automt Julin Slmnc (nd Jurrin Rot) Decemer 19, 2016 In this lecture, we will study the concept of equtions for deterministic utomt. The notes re self contined

More information

CMSC 330: Organization of Programming Languages. DFAs, and NFAs, and Regexps (Oh my!)

CMSC 330: Organization of Programming Languages. DFAs, and NFAs, and Regexps (Oh my!) CMSC 330: Orgniztion of Progrmming Lnguges DFAs, nd NFAs, nd Regexps (Oh my!) CMSC330 Spring 2018 Types of Finite Automt Deterministic Finite Automt (DFA) Exctly one sequence of steps for ech string All

More information

Languages & Automata

Languages & Automata Lnguges & Automt Dr. Lim Nughton Lnguges A lnguge is sed on n lphet which is finite set of smols such s {, } or {, } or {,..., z}. If Σ is n lphet, string over Σ is finite sequence of letters from Σ, (strings

More information

Thoery of Automata CS402

Thoery of Automata CS402 Thoery of Automt C402 Theory of Automt Tle of contents: Lecture N0. 1... 4 ummry... 4 Wht does utomt men?... 4 Introduction to lnguges... 4 Alphets... 4 trings... 4 Defining Lnguges... 5 Lecture N0. 2...

More information

First Midterm Examination

First Midterm Examination Çnky University Deprtment of Computer Engineering 203-204 Fll Semester First Midterm Exmintion ) Design DFA for ll strings over the lphet Σ = {,, c} in which there is no, no nd no cc. 2) Wht lnguge does

More information

Lexical Analysis Finite Automate

Lexical Analysis Finite Automate Lexicl Anlysis Finite Automte CMPSC 470 Lecture 04 Topics: Deterministic Finite Automt (DFA) Nondeterministic Finite Automt (NFA) Regulr Expression NFA DFA A. Finite Automt (FA) FA re grph, like trnsition

More information

CSCI 340: Computational Models. Transition Graphs. Department of Computer Science

CSCI 340: Computational Models. Transition Graphs. Department of Computer Science CSCI 340: Computtionl Models Trnsition Grphs Chpter 6 Deprtment of Computer Science Relxing Restrints on Inputs We cn uild n FA tht ccepts only the word! 5 sttes ecuse n FA cn only process one letter t

More information

12.1 Nondeterminism Nondeterministic Finite Automata. a a b ε. CS125 Lecture 12 Fall 2016

12.1 Nondeterminism Nondeterministic Finite Automata. a a b ε. CS125 Lecture 12 Fall 2016 CS125 Lecture 12 Fll 2016 12.1 Nondeterminism The ide of nondeterministic computtions is to llow our lgorithms to mke guesses, nd only require tht they ccept when the guesses re correct. For exmple, simple

More information

1.3 Regular Expressions

1.3 Regular Expressions 56 1.3 Regulr xpressions These hve n importnt role in describing ptterns in serching for strings in mny pplictions (e.g. wk, grep, Perl,...) All regulr expressions of lphbet re 1.Ønd re regulr expressions,

More information

FABER Formal Languages, Automata and Models of Computation

FABER Formal Languages, Automata and Models of Computation DVA337 FABER Forml Lnguges, Automt nd Models of Computtion Lecture 5 chool of Innovtion, Design nd Engineering Mälrdlen University 2015 1 Recp of lecture 4 y definition suset construction DFA NFA stte

More information

Tutorial Automata and formal Languages

Tutorial Automata and formal Languages Tutoril Automt nd forml Lnguges Notes for to the tutoril in the summer term 2017 Sestin Küpper, Christine Mik 8. August 2017 1 Introduction: Nottions nd sic Definitions At the eginning of the tutoril we

More information

Speech Recognition Lecture 2: Finite Automata and Finite-State Transducers

Speech Recognition Lecture 2: Finite Automata and Finite-State Transducers Speech Recognition Lecture 2: Finite Automt nd Finite-Stte Trnsducers Eugene Weinstein Google, NYU Cournt Institute eugenew@cs.nyu.edu Slide Credit: Mehryr Mohri Preliminries Finite lphet, empty string.

More information

Table of contents: Lecture N Summary... 3 What does automata mean?... 3 Introduction to languages... 3 Alphabets... 3 Strings...

Table of contents: Lecture N Summary... 3 What does automata mean?... 3 Introduction to languages... 3 Alphabets... 3 Strings... Tle of contents: Lecture N0.... 3 ummry... 3 Wht does utomt men?... 3 Introduction to lnguges... 3 Alphets... 3 trings... 3 Defining Lnguges... 4 Lecture N0. 2... 7 ummry... 7 Kleene tr Closure... 7 Recursive

More information

Non Deterministic Automata. Linz: Nondeterministic Finite Accepters, page 51

Non Deterministic Automata. Linz: Nondeterministic Finite Accepters, page 51 Non Deterministic Automt Linz: Nondeterministic Finite Accepters, pge 51 1 Nondeterministic Finite Accepter (NFA) Alphbet ={} q 1 q2 q 0 q 3 2 Nondeterministic Finite Accepter (NFA) Alphbet ={} Two choices

More information

a,b a 1 a 2 a 3 a,b 1 a,b a,b 2 3 a,b a,b a 2 a,b CS Determinisitic Finite Automata 1

a,b a 1 a 2 a 3 a,b 1 a,b a,b 2 3 a,b a,b a 2 a,b CS Determinisitic Finite Automata 1 CS4 45- Determinisitic Finite Automt -: Genertors vs. Checkers Regulr expressions re one wy to specify forml lnguge String Genertor Genertes strings in the lnguge Deterministic Finite Automt (DFA) re nother

More information

Some Theory of Computation Exercises Week 1

Some Theory of Computation Exercises Week 1 Some Theory of Computtion Exercises Week 1 Section 1 Deterministic Finite Automt Question 1.3 d d d d u q 1 q 2 q 3 q 4 q 5 d u u u u Question 1.4 Prt c - {w w hs even s nd one or two s} First we sk whether

More information

input tape head moves current state

input tape head moves current state CPS 140 - Mthemticl Foundtions of CS Dr. Susn Rodger Section: Finite Automt (Ch. 2) (lecture notes) Things to do in clss tody (Jn. 13, 2004): ffl questions on homework 1 ffl finish chpter 1 ffl Red Chpter

More information

CS415 Compilers. Lexical Analysis and. These slides are based on slides copyrighted by Keith Cooper, Ken Kennedy & Linda Torczon at Rice University

CS415 Compilers. Lexical Analysis and. These slides are based on slides copyrighted by Keith Cooper, Ken Kennedy & Linda Torczon at Rice University CS415 Compilers Lexicl Anlysis nd These slides re sed on slides copyrighted y Keith Cooper, Ken Kennedy & Lind Torczon t Rice University First Progrmming Project Instruction Scheduling Project hs een posted

More information

NON-DETERMINISTIC FSA

NON-DETERMINISTIC FSA Tw o types of non-determinism: NON-DETERMINISTIC FS () Multiple strt-sttes; strt-sttes S Q. The lnguge L(M) ={x:x tkes M from some strt-stte to some finl-stte nd ll of x is proessed}. The string x = is

More information

PART 2. REGULAR LANGUAGES, GRAMMARS AND AUTOMATA

PART 2. REGULAR LANGUAGES, GRAMMARS AND AUTOMATA PART 2. REGULAR LANGUAGES, GRAMMARS AND AUTOMATA RIGHT LINEAR LANGUAGES. Right Liner Grmmr: Rules of the form: A α B, A α A,B V N, α V T + Left Liner Grmmr: Rules of the form: A Bα, A α A,B V N, α V T

More information

Deterministic Finite-State Automata

Deterministic Finite-State Automata Deterministic Finite-Stte Automt Deepk D Souz Deprtment of Computer Science nd Automtion Indin Institute of Science, Bnglore. 12 August 2013 Outline 1 Introduction 2 Exmple DFA 1 DFA for Odd number of

More information

Kleene s Theorem. Kleene s Theorem. Kleene s Theorem. Kleene s Theorem. Kleene s Theorem. Kleene s Theorem 2/16/15

Kleene s Theorem. Kleene s Theorem. Kleene s Theorem. Kleene s Theorem. Kleene s Theorem. Kleene s Theorem 2/16/15 Models of Comput:on Lecture #8 Chpter 7 con:nued Any lnguge tht e defined y regulr expression, finite utomton, or trnsi:on grph cn e defined y ll three methods We prove this y showing tht ny lnguge defined

More information

Context-Free Grammars and Languages

Context-Free Grammars and Languages Context-Free Grmmrs nd Lnguges (Bsed on Hopcroft, Motwni nd Ullmn (2007) & Cohen (1997)) Introduction Consider n exmple sentence: A smll ct ets the fish English grmmr hs rules for constructing sentences;

More information

CS 275 Automata and Formal Language Theory

CS 275 Automata and Formal Language Theory CS 275 Automt nd Forml Lnguge Theory Course Notes Prt II: The Recognition Problem (II) Chpter II.6.: Push Down Automt Remrk: This mteril is no longer tught nd not directly exm relevnt Anton Setzer (Bsed

More information

CS 275 Automata and Formal Language Theory

CS 275 Automata and Formal Language Theory CS 275 utomt nd Forml Lnguge Theory Course Notes Prt II: The Recognition Prolem (II) Chpter II.5.: Properties of Context Free Grmmrs (14) nton Setzer (Bsed on ook drft y J. V. Tucker nd K. Stephenson)

More information

State Minimization for DFAs

State Minimization for DFAs Stte Minimiztion for DFAs Red K & S 2.7 Do Homework 10. Consider: Stte Minimiztion 4 5 Is this miniml mchine? Step (1): Get rid of unrechle sttes. Stte Minimiztion 6, Stte is unrechle. Step (2): Get rid

More information

SWEN 224 Formal Foundations of Programming WITH ANSWERS

SWEN 224 Formal Foundations of Programming WITH ANSWERS T E W H A R E W Ā N A N G A O T E Ū P O K O O T E I K A A M Ā U I VUW V I C T O R I A UNIVERSITY OF WELLINGTON Time Allowed: 3 Hours EXAMINATIONS 2011 END-OF-YEAR SWEN 224 Forml Foundtions of Progrmming

More information

CSE : Exam 3-ANSWERS, Spring 2011 Time: 50 minutes

CSE : Exam 3-ANSWERS, Spring 2011 Time: 50 minutes CSE 260-002: Exm 3-ANSWERS, Spring 20 ime: 50 minutes Nme: his exm hs 4 pges nd 0 prolems totling 00 points. his exm is closed ook nd closed notes.. Wrshll s lgorithm for trnsitive closure computtion is

More information

Exercises Chapter 1. Exercise 1.1. Let Σ be an alphabet. Prove wv = w + v for all strings w and v.

Exercises Chapter 1. Exercise 1.1. Let Σ be an alphabet. Prove wv = w + v for all strings w and v. 1 Exercises Chpter 1 Exercise 1.1. Let Σ e n lphet. Prove wv = w + v for ll strings w nd v. Prove # (wv) = # (w)+# (v) for every symol Σ nd every string w,v Σ. Exercise 1.2. Let w 1,w 2,...,w k e k strings,

More information

1.4 Nonregular Languages

1.4 Nonregular Languages 74 1.4 Nonregulr Lnguges The number of forml lnguges over ny lphbet (= decision/recognition problems) is uncountble On the other hnd, the number of regulr expressions (= strings) is countble Hence, ll

More information

CS:4330 Theory of Computation Spring Regular Languages. Equivalences between Finite automata and REs. Haniel Barbosa

CS:4330 Theory of Computation Spring Regular Languages. Equivalences between Finite automata and REs. Haniel Barbosa CS:4330 Theory of Computtion Spring 208 Regulr Lnguges Equivlences between Finite utomt nd REs Hniel Brbos Redings for this lecture Chpter of [Sipser 996], 3rd edition. Section.3. Finite utomt nd regulr

More information